For simplicity let's talk about $p$-localizations of spaces for a fixed prime $p$. Every space $X$ has a well-defined $p$-localization which can be constructed by the small object argument and which becomes a fibrant replacement in the $p$-local model structure on the category of spaces. It is well-known that nilpotent spaces have nice enough Postnikov towers and we can localize such spaces by taking the Postnikov tower, localizing step by step and putting it back together by taking the limit of the resulting tower of fibrations. My question is:

Is there an example of a non-nilpotent space $X$ whose $p$-localization we can explicitly describe?

I leave the meaning of "explicitly" ambiguous. I would be interested in any construction not using the small object argument.

Here's my stab at a possible example. For a group $G$ we define its lower central series by setting $G_0 = G$ and $G_{n + 1} = [G_n, G]$ and we can also continue transfinitely by setting $G_\beta = \bigcap_{\alpha < \beta} G_\alpha$ for limit ordinals $\beta$. The group $G$ is nilpotent if this construction terminates at the trivial subgroup at a finite stage. It is called hypocentral if it terminates at the trivial subgroup at some not necessarily finite stage. According to Wikipedia it is a result of Malcev that there are hypocentral groups with arbitrarily long lower central series.

If we start with a hypocentral group $G$ and convert its lower central series into a (transfinite) tower of fibrations (whose limit is a $K(G, 1)$), $p$-localize it step by step and take the limit of the resulting tower of fibrations, do we obtain the $p$-localization of $K(G, 1)$?

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As you know extending $P$-localization functors to non-nilpotent spaces is a very delicate matter. You can do it by homotopical localization techniques. if you do it in this way you get functorial localization which work for non-nilpotent spaces "but" it is the aim of your question it is not "very explicit".

Now let me be very naive and let us go back to D. Sullivan's MIT notes "Geometric topology, localization, periodicity and Galois symmetry" chapter 2 (these notes are really beautiful to read).
D. Sullivan first explains how to do $P$-localization when we have a CW-complex, of course he also explains that this CW-procedure gives a "nice localization" when the CW-complex is 1-connected (in that case it is our modern $P$-localization). He proceeds on the skeleton of the CW-complex, localizing cell by cell.
Thus you can apply Sullivan's telescope construction to a circle and then to a wedge of circles and any CW-complex you want. The starting point being the localization of spheres through a telescope of maps $S^k\stackrel{\times p}{\rightarrow }S^k$ (you invert degree $p$-maps).

The point is that you can argue that this naive construction on non-nilpotent CW-complexes is not what you expected as a $P$-localization. D. Sullivan also explains how to do it when we have a Postnikov tower.

Let me add that there is another interesting way to $P$-localize which is due to C. Casacuberta and G. Peschke:
"Localizing with respect to self-maps of the circle" Trans AMS 339 (1993) 117 – 140.

This $P$-localization is as "explicit" as a homological Bousfield's localization but the homotopy related to it is very interesting. In that setting, a space is $P$-local if and only if the $p$-power map on the loop space $\Omega X$ is a self homotopy equivalence if $p\notin P$.